Non-projective quantum measurement of solid-state qubits: Bayesian formalism (what is inside collapse)

Transcription

1 Kending, Taiwan, 01/16/11 Non-projective quantum measurement of solid-state qubits: Bayesian formalism (what is inside collapse) Outline: Very long introduction (incl. EPR, solid-state qubits) Basic Bayesian formalism for quantum measurement and its derivations Non-ideal detectors Bayesian formalism in circuit QED setup Acknowledgements Many useful discussions and collaborations

2 Quantum mechanics is weird Niels Bohr: If you are not confused by quantum physics then you haven t really understood it Richard Feynman: I think I can safely say that nobody understands quantum mechanics Weirdest part is quantum measurement

3 Quantum mechanics = Schrödinger equation (evolution) + collapse postulate (measurement) 1) Probability of measurement result p r = ψ ψ r ) Wavefunction after measurement = ψ r State collapse follows from common sense Does not follow from Schrödinger equation (contradicts, random vs. deterministic, philosophy ) Collapse postulate is controversial since 190s (needs an observer, contradicts causality) Our focus: what is inside collapse, but first discuss EPR

4 Einstein-Podolsky-Rosen (EPR) paradox Phys. Rev., 1935 In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. ψ ( x, x ) = ψ ( x ) u ( x ) 1 n n n 1 1 = 1 δ 1 ψ ( x, x ) exp[( i/ )( x x ) p] dp ~ ( x x ) x1 x Bohr s reply (Phys. Rev., 1935) (except in footnotes) It is shown that a certain criterion of physical reality formulated by A. Einstein, B. Podolsky and N. Rosen contains an essential ambiguity when it is applied to quantum phenomena. (nowadays we call it entangled state) Measurement of particle 1 cannot affect particle, while QM says it affects (contradicts causality) => Quantum mechanics is incomplete (seven pages, one formula: Δp Δq ~ h) Very crudely: No need to understand QM, just use the result

5 a Bell s inequality (John Bell, 1964) (setup by David Bohm) b Advantage: choice of meas. directions ψ = 1 ( 1 ) 1 = b Perfect anticorrelation of results for same meas. directions, a Is it possible to explain the QM result assuming local realism and hidden variables (without superluminal collapse)? No!!! Assume: Aa (, λ) =± 1, Bb (, λ) =± 1, (deterministic result with perfect anticorr. for ( aa, ) hidden variable λ) Then: P ( ab, ) Pac (, ) 1 + Pbc (, ) where P P( ++ ) + P( ) P( + ) P( + ) QM: P( ab, ) = ab i For 0, 90, and 45 : violation! Experiment (Aspect et al., 198; photons instead of spins, CHSH): yes, spooky action-at-a-distance 5/44

6 CHSH paper (Clauser, Horne, Shimony, Holt, 1969) a or a b or b 4 experiments instead of 3 In CHSH perfect anticorrelation not required practical S, where S = P( a, b) - P( a, b') + P( a', b) + P( a', b') P p( ++ ) + p( --)- p( +-) - p( -+) (Aspect s version) Maximum violation by QM: S = ± P(a,b) =-cos(a,b) a b a b a S = S =- b b a a=0, a =70, a=0, a =90, b=135, b =45 b=45, b =135 Problem with original Bell ineq.: need perfect anticorrelation for same directions not practical! Easy derivation: a a b b S Probab. by averaging

7 What about causality? Actually, not too bad: you cannot transmit your own information choosing a particular measurement direction a a or Result of the other measurement does not depend on direction a Randomness saves causality Collapse is still instantaneous: OK, just our recipe, not an objective reality, not a physical process Consequence of causality: No-cloning theorem Proof: Wootters-Zurek, 198; Dieks, 198; Yurke You cannot copy an unknown quantum state Otherwise get information on direction a (and causality violated) Application: quantum cryptography Information is an important concept in quantum mechanics

8 Quantum measurement in solid-state systems No violation of locality too small distances However, interesting issue of continuous measurement (weak coupling, noise gradual collapse) Same origin of paradoxes as in EPR (Schr. Eq. not enough) Starting point: qubit detector I(t), noise S What happens to a solid-state qubit (two-level system) during its continuous measurement by a detector?

9 Superconducting charge qubit Y. Nakamura, Yu. Pashkin, and J.S. Tsai (Nature, 1998) E - J ˆ ( e) H = ( nˆ -ng ) C ( n n+ 1 + n+ 1 n ) e Vg island Josephson junction n g Single Cooper pair box n n+1 Vion et al. (Devoret s group); Science, 00 Q-factor of Ramsey oscillations = 5,000 Quantum coherent (Rabi) oscillations Δt (ps) E J n: number of Cooper pairs on the island

10 Charge qubits with SET readout Vg V I(t) Duty, Gunnarsson, Bladh, Delsing, PRB 004 Guillaume et al. (Echternach s group), PRB 004 e Cooper-pair box measured by singleelectron transistor (rf-set) Setup can be used for continuous measurements All results are averaged over many measurements (not single-shot ) 10/44

11 Some other superconducting qubits Flux qubit Mooij et al. (Delft) Phase qubit J. Martinis et al. (UCSB and NIST) Charge qubit with circuit QED R. Schoelkopf et al. (Yale)

12 Some other superconducting qubits Flux qubit J. Clarke et al. (Berkeley) Quantronium qubit I. Siddiqi, R. Schoelkopf, M. Devoret, et al. (Yale)

13 Semiconductor (double-dot) qubit T. Hayashi et al. (NTT), PRL 003 Detector is not separated from qubit, also possible to use a separate detector Rabi oscillations

14 Some other semiconductor qubits Spin qubit (QPC meas.) C. Marcus et al. (Harvard) Spin qubit L. Kouwenhoven et al. (Delft) Double-dot qubit Gorman, Hasko, Williams (Cambridge)

15 Which-path detector experiment Buks, Schuster, Heiblum, Mahalu, and Umansky, Nature 1998 A-B loop I(t) V QPC detector Dephasing rate: ev ( ΔT ) ( I ) Γ= = Δ h T(1 T) S 4 I ΔI detector response, S I shot noise The larger noise, the smaller dephasing!!! 15/44 Theory: Aleiner, Wingreen, and Meir, PRL 1997 (ΔI) /4S I ~ rate of information flow

16 What is the evolution due to measurement? (What is inside collapse?) controversial for last 80 years, many wrong answers, many correct answers solid-state systems are more natural to answer this question Various approaches to non-projective (weak, continuous, partial, generalized, etc.) quantum measurements Names: Davies, Kraus, Holevo, Mensky, Caves, Knight, Plenio, Walls, Carmichael, Milburn, Wiseman, Gisin, Percival, Belavkin, etc. (very incomplete list) Key words: POVM, restricted path integral, quantum trajectories, quantum filtering, quantum jumps, stochastic master equation, etc. Our limited scope: (simplest system, experimental setups) solid-state qubit detector I(t), noise S

17 Typical setup: double-quantum-dot (DQD) qubit + quantum point contact (QPC) detector 1Ò Ò H e I(t) 1Ò Ò Ò 1Ò I(t) Gurvitz, 1997 H = H QB + H DET + H INT ε H = σ + QB H σ Z X ΔI I() t = I0 + z() t + ξ () t const + signal + noise Two levels of average detector current: I 1 for qubit state 1, I for Response: ΔI = I 1 I Detector noise: white, spectral density S I For low-transparency QPC + SI = ei DET = l l l r r r + l r l, r r l + l r H Eaa Eaa Taa ( aa ) INT = Δ, 1 1 lr r l + l r H T ( cc cc )( aa aa )

18 What happens to a qubit state during measurement? Start with density matrix evolution due to measurement only (H=ε=0 ) Orthodox answer Decoherence answer exp( Γt) exp( Γt) > or >, depending on the result no measurement result! (ensemble averaged) Decoherence has nothing to do with collapse! applicable for: single quant. system continuous meas. Orthodox yes no Decoherence (ensemble) no yes Bayesian, POVM, quant. traject., etc. yes yes Bayesian (POVM, quant. traj., etc.) formalism describes gradual collapse of a single quantum system, taking into account measurement result

19 Bayesian formalism for DQD-QPC system H QB =0 1Ò Ò H e I(t) Qubit evolution due to measurement (quantum back-action): ψ () t = α() 1 t + β() t or ρ ij () t Bayes rule (1763, Laplace-181): posterior probability P( A ) (res ) ( res) i P A P A i i = P( A ) P(res A ) k 1) α(t) and β(t) evolve as probabilities, i.e. according to the Bayes rule (same for ρ ii ) ) phases of α(t) and β(t) do not change (no dephasing!), ρ ij /(ρ ii ρ jj ) 1/ = const k k 1 τ τ I() t dt 0 So simple because: (A.K., 1998) prior measured probab. likelihood I 1 I 1) no entaglement at large QPC voltage ) QPC happens to be an ideal detector 3) no Hamiltonian evolution of the qubit

20 Quantum Bayes theorem (ideal detector assumed) 1> H > e I(t) H = ε = 0 ( frozen qubit) After the measurement during time τ, the probabilities should be updated using the standard Bayes formula: Quantum Bayes formulas: 0/44 P ρ _ Measurement (during time τ): I actual D 1/ D 1/ I 1 I _ I Initial state: τ ρ11(0) ρ1(0) ρ1(0) ρ(0) 1 I I() t dt τ 0 P( I, τ ) = ρ11(0) P1 ( I, τ) + ρ(0) P ( I, τ) 1 P i ( I, τ ) = exp[ ( I I ) / ], i D π D D= S I I I S I I / τ, 1 i, τ I / i P( B ) ( ) ( ) i P A B P B i i A = P( Bk) P( A Bk) ρ11(0) exp[ -( I -I1) / D] 11( τ) = ρ ρ - - (0) exp[ ( I I ) / D] (0) exp[ ( I I ) / D] ρ ( τ) ρ (0) =, ρ( τ) = 1 -ρ11( τ) [ ρ ( τ) ρ ( τ)] [ ρ (0) ρ (0)] 1 1 1/ 1/ k

21 H e Bayesian formalism for a single qubit I(t) e Vg V I(t) Time derivative of the quantum Bayes rule Add unitary evolution of the qubit Add decoherence γ (if any) H ΔI ρ =- ρ =- Im ρ + ρ ρ [ It ()-I ] SI H ΔI ρ = iερ + i ( ρ -ρ ) + ρ ( ρ -ρ ) [ It ()-I ]-γ ρ SI (A.K., 1998) ΔI=I 1 -I, I 0 =(I 1 +I )/, S I detector noise ΔI γ = 0 for QPC For simulations: I = I0 + ( ρ11 ρ) + ξ noise S Evolution of qubit wavefunction can be monitored if γ=0 (quantum-limited) Natural generalizations: add classical back-action entangled qubits ξ = S I

22 H e I(t) e Vg V I(t) ΔI detector response, S I detector noise Relation to conventional master equation ΔI ρ =- ρ =- H Im ρ + ρ ρ [ It ()-I ] SI ΔI ρ = iερ + ih( ρ -ρ ) + ρ ( ρ -ρ ) [ It ()-I ]-γ ρ SI Averaging over result I(t) leads to conventional master equation: ρ =- ρ / dt =-H Imρ 11 1 ρ 1 = iερ1 + ih( ρ11 -ρ)-γρ1 Γ ensemble decoherence, Γ = γ + Δ ( I) /4S I Ensemble averaging includes averaging over measurement result ( ΔI) / 4S quantum decoherence Quantum efficiency: η = I = = 1 Γ total total = 1

23 Assumptions needed for the Bayesian formalism: Detector voltage is much larger than the qubit energies involved ev >> ÑΩ, ev >> ÑΓ, Ñ/eV << (1/Ω,1/Γ), Ω=(4H +ε ) 1/ (no coherence in the detector, classical output, Markovian approximation) Simpler if weak response, ΔI << I 0, (coupling C ~ Γ/Ω is arbitrary) Derivations: 1) logical : via correspondence principle and comparison with decoherence approach (A.K., 1998) ) microscopic : Schr. eq. + collapse of the detector (A.K., 000) n ρ ij () t nt ( k ) qubit detector pointer quantum interaction quantum frequent collapse classical information n number of electrons passed through detector 3) from quantum trajectory formalism developed for quantum optics (Goan-Milburn, 001; also: Wiseman, Sun, Oxtoby, etc.) 4) from POVM formalism (Jordan-A.K., 006) 5) from Keldysh formalism (Wei-Nazarov, 007)

24 Informational derivation of the Bayesian formalism Step 1. Assume H = ε = 0 ( frozen qubit). Since ρ 1 is not involved, evolution of ρ 11 and ρ should be the same as in the classical case, i.e. Bayes formula (correspondence principle). (A.K., 1998) Step. Assume H = ε = 0 and pure initial state: ρ 1 (0) = [ρ 11 (0) ρ (0)] 1/. For any realization ρ 1 (t) [ρ 11 (t) ρ (t)] 1/. Then averaging over realizations gives ρ 1 av (t) ρ 1 av (0) exp[æ(δi /4S I ) t]. Compare with conventional (ensemble) result (Gurvitz-1997, Aleiner et al.-97) for QPC: ρ 1 av (t)=ρ 1 av (0) exp[æ(δi /4S I ) t]. Exactly the upper bound! Therefore, pure state remains pure: ρ 1 (t)=[ρ 11 (t) ρ (t)] 1/. Step 3. Account of a mixed initial state Result: the degree of purity ρ 1 (t)/[ρ 11 (t) ρ (t)] 1/ is conserved. Step 4. Add qubit evolution due to H and ε. Step 5. Add extra dephasing due to detector nonideality (i.e., for SET).

25 Schrödinger evolution of qubit + detector for a low-t QPC as a detector (Gurvitz, 1997) d I I H ρ ρ ρ dt e e ρ d I I H ρ ρ ρ dt e e ρ n 1 n 1 n 1 n 11 = Im 1 n n n 1 n = + + Im 1 d ε H I + I II ρ ρ ρ ρ ρ ρ dt e e n n n n 1 n 1 n 1 1 = i 1 + i ( 11 ) If H = ε =0, this leads to 5/44 Microscopic derivation of the Bayesian formalism ρ ρ n ρ ij () t nt ( k ) qubit detector pointer quantum interaction ρ frequent collapse n number of electrons passed through detector Detector collapse at t = t k Particular n k is chosen at t k n n Pn ( ) = ρ11( tk ) + ρ( t k ) n ij tk + n, nk ij tk + nk ρij ( tk ) ij ( tk + 0) = nk nk ρ11 tk + ρ ρ ( 0) = δ ρ ( 0) (0) P ( n) ρ (0) P ( n) ρ ( ) ( t ) () t =, ρ() t = ρ ρ ρ ρ 1/ n [ ρ11( t) ρ( t)] ( It/ ) 1() 1(0) 1/, ( ) i e t = ρ Pi n = exp( Iit/ e), [ ρ11(0) ρ(0)] n! (0) P ( n) (0) P ( n) (0) P ( n) (0) P ( n) which are exactly quantum Bayes formulas (A.K., 000) classical information k

26 Derivation via POVM Quantum measurement in POVM formalism (Nielsen-Chuang, p. 85,100): Measurement (Kraus) operator M Mrρ M rψ r ψ or ρ M r (any linear operator in H.S.) : Mrψ Tr( Mr ρ Mr) Probability : Pr = Mrψ or Pr = Tr( Mr ρ Mr) Completeness : r r = (People often prefer linear evolution 1 and non-normalized states) reflected (r 1, ) incident electron 1Ò Ò transmitted r M M (Jordan, A.K., 006) For each incident in ( α 1 + β ) α( r1 L + t1 R ) 1 electron: + β ( r L + t R ) r1 0 t1 0 Mrefl =, Mtrans = (t 1, ) 0 r 0 t For many incident electrons Bayesian formalism Relation between POVM and quantum Bayesian formalism: decomposition r = r r r M U M M (almost equivalent) unitary Bayes

27 Where POVM measurement comes from Initial state ψ = c k in Interaction with ancilla la, k 0 U l a k kla, k (U comes from unitary transformation in combined Hilbert space system+ancilla) ψ in 0 cu k k, la l a = l cu k k, la a kla,, l ka, Project ancilla onto r = r a a a 1 1 * in cu k k, la r c k ru a k, la Norm l k, a Norm l k a ψ 0 l a r = l r So, as a result: ancilla 0 r system k c k k kl, M c rlk, k Norm l system i.e. ψ ancilla M ψ r in in Norm projective measurement M rlk,

28 Ito form Quantum trajectory formalism for the same system Goan, Milburn, Wiseman, Sun, 000 Goan, Milburn, 001 ξ(t) -white noise Looks different, but equivalent to Bayesian formalism

29 Stratonovich and Ito forms for nonlinear stochastic differential equations Definitions of the derivative: df ( t) f ( t +Δt/ ) f ( t Δt / ) = limδ t 0 dt Δt df () t f ( t +Δt) f () t = limδ t 0 (Ito) dt Δt Why matters? But if df = ξ dt Simple translation rule: (Stratonovich) Usually ( f + df ) f + f df, ( df ) df S ξ (white noise ξ), then ( df ) = ξ dt dt d x () t i = G ( x,) t i + F ( x,) t i ξ () t dt d S Fi ( x, t) x i() t G i( x,) t F i( x,) t () t ξ = + ξ + F k( x,) t dt 4 dx k (Stratonovich) k (Ito) Advantage of Stratonovich: usual calculus rules (intuition) Advantage of Ito: simple averaging

30 Monte Carlo Methods for calculations Ideologically simplest In many cases most efficient Idea: use finite time step Δt - find probability distribution for I(Δt) - pick a random number for I(Δt) do quantum Bayesian update Analytics (or non-random numerics) Be very careful about Ito-Stratonovich issue Use Stratonovich form for derivations (derivatives, etc.) Convert into Ito for averaging over noise Very good idea to compare with Monte Carlo and/or check second order terms in dt 30/44

31 Fundamental limit for ensemble decoherence ensemble decoherence rate Γ = (ΔI) /4S I + γ γ 0 γ ( ΔI) /4SI 1 η = - = Γ Γ single-qubit decoherence ~ information flow [bit/s] Γ (ΔI) /4S I detector ideality (quantum efficiency) η 100% Translated into energy sensitivity: (ε O ε BA ) 1/ / ε O, ε BA : sensitivities [J/Hz] limited by output noise and back-action Γτ m A.K., 1998, 000 S. Pilgram et al., 00 A. Clerk et al., 00 D. Averin, 000,003 Known since 1980s (Caves, Clarke, Likharev, Zorin, Vorontsov, Khalili, etc.) (ε O ε BA - ε O,BA ) 1/ / Γ (ΔI) /4S I + K S I /4 measurement time (S/N=1) τ m = SI /( ΔI ) 1 Danilov, Likharev, Zorin, 1983

32 Two ways to think about a non-ideal detector (η<1) qubit ( ΔI ) Γ Σ = 4S 0 ideal detector S 0 S 1 + noise SΣ = S0 + S1 I(t) η = ( ΔI) /4S Γ Σ Σ dephasing noise Γ =Γ +Γ Σ 0 1 These ways are equivalent (same results for any expt.) fi matter of convenience Γ 1 qubit ( ΔI ) Γ 0 = 4 S Σ ideal detector S Σ I(t)

33 Nonideal detectors with input-output noise correlation classical noise affecting ε qubit (ε, H) signal quantum backaction noise classical noise affecting ε ξ (t) = Aξ 1 (t) ideal detector ξ 3 (t) fully correlated Id (t) S0 classical current S1 ξ1 (t) + detector I(t) S0+S1 d d ΔI ρ = ρ = HIm ρ + ρ ρ [ I( t) I ] dt dt S I A.K., 00 AS1+ θ S K = 0, SI = S + S S 0 1 d ΔI ρ = iερ + ih( ρ ρ ) + ρ ( ρ ρ ) [ I( t) I ] + ik[ I( t) I ] ρ γ ρ dt S I quantum efficiency : γ Δ I SI + K SI 1 η = - = Γ ( ) /4 /4 Γ or I K correlation between output and ε backaction noises η = ( ΔI) /4S I Γ

34 1Ò Ò A simple general form for a broadband linear detector (QPC, SET, etc.) Δ I = I1-I noise S I 1 τ I I() t dt τ 0 D= S /τ I I 1 I H qb = 0 quantum backaction (non-unitary, Ò spooky, unphysical ) 1Ò ρ11( τ) ρ11(0) exp[ -( I -I1) / D] = no self-evolution I(t) ρ( τ) ρ(0) exp[ -( I -I) / D] of qubit assumed ρ11( τ) ρ( τ) ρ1( τ) = ρ1(0) exp( ikiτ)exp( γτ) ρ11(0) ρ(0) decoherence classical backaction (unitary) Example of classical ( physical ) backaction: Each electron passed through QPC rotates qubit (sensitivity of tunneling phase for an asymmetric barrier) * arg( T ΔT) 0 + DET = l l l r r r + l r l, r r l + l r INT = Δ (, 1 1 )( r l + l r) lr H Eaa Eaa Taa ( aa) H T cc cc aa aa

35 e A simple general form for a broadband linear detector (QPC, SET, etc.) Vg V Δ I = I1-I noise S I 1 τ I I() t dt τ 0 D= S /τ Γ L I V I(t) H qb = I(t) 0 ϕ (t) Γ R 0 quantum backaction (non-unitary, spooky, unphysical ) ρ ( τ) ρ (0) exp[ ( I I ) / D] = ρ ( τ) ρ (0) exp[ ( I I ) / D] ρ11( τ) ρ( τ) ρ1( τ) = ρ1(0) exp( ikiτ)exp( γτ) ρ11(0) ρ(0) decoherence classical backaction (unitary) Another example of classical backaction: no self-evolution of qubit assumed Correlation between voltage and current noises in SET SIϕ (0) ΓL ΓR SI ϕ = SII (0) Sϕϕ (0) ( Γ L + Γ R) A.K., 1994 (easy to understand when Γ L <<Γ R ) 35/44

36 Narrowband linear measurement Difference from broadband: two quadratures System: qubit in cqed setup + parametric amplifier microwave generator resonator qubit (transmon) pumps paramp quantum signal ( quadratures) ω d ω r ω a ω b mixer Q(t) I(t) Yale output (two quadratures) Paramp traditionally discussed in terms of noise temperature θ 0 ω θ for phase-sensitive (degenerate, homodyne) paramp for phase-preserving (non-degenerate, heterodyne) paramp Haus, Mullen, 196 Ackn.: Likharev, Giffard, 1976 Devoret We will discuss it in terms of qubit evolution due to measurement

37 microwave generator resonator qubit (transmon) pumps ω d ω r ω a ω b paramp mixer Q(t) I(t) output (two quadratures) Simplest case ωqb σ z + ω r χ σ z H = a a + a a ωr Q κout χ κ ωd = κ max( Γ, Ω ) Blais et al., 004 Gambetta et al., 006, 008 R (dispersive) (Markovian, bad cavity ) = κ (everything collected; i.e. reflection) r (weak response) = ω (center of resonance, only phase change if transmission) Ò assume everything most ideal carries information about qubit (σ z ) (quantum back-action) cos( ω d t ) 1Ò sin( ω d t ) carries information about fluctuating photon number in the resonator (classical back-action)

38 μwave gen. resonator qubit pumps ω d ω r ω a ω b paramp mixer Phase-sensitive (degenerate) paramp pumps ω a +ω b =ω d Q(t) I(t) cos( ω d t ) 1Ò Ò ϕ quadrature cos(ω d t +ϕ) is amplified, quadrature sin(ω d t +ϕ) is suppressed Assume I(t) measures cos(ω d t +ϕ), then Q(t) not needed amplified sin( ω d t ) get some information (~cos ϕ) about qubit state and some information (~sin ϕ) about photon fluctuations ρgg( τ) ρgg(0) exp[ -( I -Ig ) / D] 1 τ I I() t dt = τ D= S 0 I /τ ρee( τ) ρee(0) exp[ -( I -Ie ) / D] ΔI I ρgg( τ) ρee( τ) g Ie =ΔIcosϕ K = sinϕ ρge( τ) = ρge(0) exp( ikiτ) S I ρgg(0) ρee(0) ( ΔI cos ϕ ) SI ΔI 8χ n Γ = + K = = unitary 4S 4 4 (rotating frame) I SI κ Same as for QPC/SET, but trade-off (ϕ) between quantum & classical back-actions I(t)

39 μwave gen. resonator qubit pumps ω d ω r ω a ω b paramp mixer Q(t) I(t) Phase-preserving (nondegenerate) paramp pumps ω a +ω b =(ω d +δω) ϕ = δω t Now information in both I(t) and Q(t). Small δω can follow ϕ(t) Large δω (>>Γ) averaging over ϕ (phase-preserving) ρgg( τ) ρgg(0) exp[ -( I -Ig ) / D] = ρ ( τ) ρ (0) exp[ -( I -I ) / D] ee ee e ρgg( τ) ρee( τ) ρge( τ) = ρge(0) exp( ikqτ) ρ (0) ρ (0) gg (rotating frame) ee I τ 1 1 I() t dt Q () D = 0 Q t dt 0 τ I g ΔI ΔI Ie = K = S ΔI ΔI 8χ n Γ = + = 8S 8S κ I I Equal contributions to ensemble dephasing from quantum & classical back-actions I I(t) 1Ò 0Ò cos( ω d t ) ϕ sin( ω d t ) Q(t) δω t Choose I(t) cos(ω d t) (qubit information) Q(t) sin(ω d t) (photon fluct. info) τ τ S I τ

40 Why not just use Schrödinger equation for the whole system? qubit detector information Impossible in principle! 40/44 Technical reason: Outgoing information (measurement result) makes it an open system Philosophical reason: Random measurement result, but deterministic Schrödinger equation Einstein: God does not play dice Heisenberg: unavoidable quantum-classical boundary

41 Measurement vs. decoherence Widely accepted point of view: measurement = decoherence (environment) Is it true? Yes, if not interested in information from detector (ensemble-averaged evolution) No, if take into account measurement result (single quantum system) Measurement result obviously gives us more information about the measured system, so we know its quantum state better (ideally, a pure state instead of a mixed state)

42 Can we verify the Bayesian formalism experimentally? Direct way: prepare partial measur. control (rotation) projective measur. However, difficult: bandwidth, control, efficiency (expt. realized only for supercond. phase qubits) A.K.,1998 Tricks are needed for real experiments

43 Experimental predictions and proposals from Bayesian formalism Direct experimental verification (1998) Measured spectrum of Rabi oscillations (1999, 000, 00) Bell-type correlation experiment (000) Quantum feedback control of a qubit (001, 004, 009) Entanglement by measurement (00) Measurement by a quadratic detector (003) Squeezing of a nanomechanical resonator (004) Violation of Leggett-Garg inequality (005) Partial collapse of a phase qubit (005) Undoing of a weak measurement (006, 008) Decoherence suppression by uncollapsing (010) Persistent Rabi revealed in noise (010) 3 solid-state experiments realized so far

44 Conclusions Quantum measurement is the most controversial and fascinating part of quantum mechanics It is easy to see what is inside collapse: simple Bayesian formalism works for many solid-state setups Classical information plays a very important part in quantum measurement Measurement backaction necessarily has a spooky part ( unphysical, informational, without a mechanism); it may also have a classical part (with a physically understandable mechanism) 44/44